Wow, good advice you're getting!
Actually, I strap my car seat to a luggage cart and it works just as well as that expensive GogoKidz thingy. My toddlers could ride in it when tired. Use an extra bungee if necessary.
You can either purchase a seat or take your chances of getting an extra place if the flight isn't full. The latter is a gamble but call the airlines off hours and ask (try early in the morning since you're in the eastern half of the country) before work hours in most time zones.
Don't be suckered into "children's discounts". Just do the math and get the cheapest seats! I've had it work out cheaper to pay the same (lower) rate for all of us.
I'm kind of new to ticket-purchasing myself, since I was with the airlines for so long, with discounted standby tickets.
-Check with travel agents, email them to save time
-Check low-priced websites like Expedia
-Check the airlines own websites
and the one I just learned
-Call the airline directly. I got the best deal by far just a few weeks ago for our Xmas tickets!
Here are my non-commercial flying tips;http://flyingwithchildren1.blogspot.com